Knowledge Spaces with Arbitrarily High Rank

Abstract The class of probabilistic belief spaces (Harsanyi, 1967–68, Man. Sci. , 14, 159–182, 320–324, 486–502) contains a universal space , into which every other belief space can be mapped in a unique way by a belief morphism. We show that there is no analogous universal space in the class of knowledge spaces. To show this we define the rank of a knowledge space, which is the ordinality of the most complicated descriptions of knowledge in the space. We then show that a knowledge space can be mapped by a knowledge morphism only to spaces of higher or equal rank. We construct knowledge spaces for arbitrarily high rank, demonstrating that there is no universal space. Journal of Economic Literature Classification Numbers: D80, D82.